Triangle Calculator

How GPS Knows Where You Are — Triangulation From Ancient Greece to Satellites

Around 240 BCE, Eratosthenes measured the circumference of the Earth using two sticks, the angle of shadows, and the Law of Sines. The insight: if you know one side of a triangle and two angles, you can find every other dimension. GPS positioning today works the same way — your phone receives timing signals from multiple satellites, forms triangles in 3D space using known satellite positions and signal differences, then solves those triangles to fix your location. Every navigation system on Earth is running triangle calculations continuously, from ancient surveying to modern satellite geometry.

This free triangle calculator solves any triangle given three independent measurements — SSS (three sides), SAS (two sides and the included angle), AAS (two angles and one side), or Right Triangle (two legs). All missing sides, angles, area and perimeter are computed instantly. The triangle type — Equilateral, Isosceles, Scalene, or Right — is identified automatically.

Heron's Formula — Area When You Have No Height

The standard area formula ½ × base × height requires the perpendicular height, which is almost never given directly in real surveying or construction problems. Heron of Alexandria devised a formula using only the three side lengths: compute the semi-perimeter s = (a+b+c)/2, then Area = √(s(s−a)(s−b)(s−c)). No angles needed. For a 3-4-5 triangle: s = 6, Area = √(6×3×2×1) = √36 = 6 — confirmed by ½ × 3 × 4 = 6. SSS mode in this calculator uses Heron's formula internally.

The Law of Cosines — c² = a² + b² − 2ab·cos(C) — generalises the Pythagorean theorem. When angle C = 90°, cos(90°) = 0 and the formula reduces to c² = a² + b². For any other angle it adjusts the result: acute angles make the third side shorter, obtuse angles make it longer. This is what SAS mode uses to find the unknown side when two sides and the angle between them are known. The Law of Sines — a/sin(A) = b/sin(B) = c/sin(C) — handles the remaining configurations in AAS mode.

Triangle Inequality: Why Some Side Combinations Are Impossible

Not every set of three positive numbers forms a valid triangle. Each side must be strictly less than the sum of the other two. Sides 1, 2, 10 fail: 1 + 2 = 3 < 10, so those three lengths can never close into a triangle — the two short sides cannot reach each other across the long one. The calculator checks this automatically and shows an error immediately rather than producing a nonsensical result. This validation matters in construction and engineering — entering a structurally impossible configuration needs to fail clearly, not silently.

Computer graphics uses triangles as its universal primitive — every 3D mesh, whether a game character, a product render, or a medical scan reconstruction, is a surface made entirely of triangles. Lighting, shadow casting, collision detection and texture mapping all reduce to triangle geometry. Architecture uses triangle calculations for roof pitch, rafter length, truss design and verifying right angles. For right-triangle-specific problems, the Pythagorean Theorem Calculator provides a cleaner focused interface with step-by-step working. For trig function evaluation with precise DEG/RAD control, use the Scientific Calculator. All calculations here run entirely in your browser.

✓Verified by ToollyX Team · Last updated June 2026